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On a Generalisation of Finite T-Groups

Communications in Mathematics and Statistics volume 10, pages 153–162 (2022)Cite this article

Abstract

Let \(\sigma =\{\sigma _i |i\in I\}\) be some partition of all primes \({\mathbb {P}}\) and G a finite group. A subgroup H of G is said to be \(\sigma \)-subnormal in G if there exists a subgroup chain \(H=H_0\le H_1\le \cdots \le H_n=G\) such that either \(H_{i-1}\) is normal in \(H_i\) or \(H_i/(H_{i-1})_{H_i}\) is a finite \(\sigma _j\)-group for some \(j \in I\) for \(i = 1, \ldots , n\). We call a finite group G a \(T_{\sigma }\)-group if every \(\sigma \)-subnormal subgroup is normal in G. In this paper, we analyse the structure of the \(T_{\sigma }\)-groups and give some characterisations of the \(T_{\sigma }\)-groups.

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Chi Zhang

  2. School of Science, Hainan University, Haikou, Hainan, 570228, People’s Republic of China

    Wenbin Guo

  3. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Wenbin Guo

  4. School of Science, Hainan University, Haikou, 570228, People’s Republic of China

    A-Ming Liu

Authors
  1. Chi Zhang
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  2. Wenbin Guo
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  3. A-Ming Liu
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Corresponding author

Correspondence to Chi Zhang.

Additional information

Research was supported by the Fundamental Research Funds for the Central Universities (No. 2020QN20) and NSFC of China(No. 12001526)

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Zhang, C., Guo, W. & Liu, AM. On a Generalisation of Finite T-Groups. Commun. Math. Stat. 10, 153–162 (2022). https://doi.org/10.1007/s40304-021-00240-z

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  • DOI: https://doi.org/10.1007/s40304-021-00240-z

Keywords

  • Finite groups
  • \(\sigma \)-groups
  • Generalised T-groups
  • \(\sigma \)-subnormal
  • The condition \({\mathfrak {R}}_{\sigma _i}\)

Mathematics Subject Classification

  • 20D10
  • 20D15
  • 20D20
  • 20D35